Board Paper of Class 10 2020 Maths (Standard) Delhi(Set 3)  Solutions
General Instructions :
(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory:
(ii) Section A : Q. No. 1 to 20 comprises of 20 questions of one mark each.
(iii) Section B : Q. No. 21 to 26 comprises of 6 questions of two marks each.
(iv) Section C: Q. No. 27 to 34 comprises of 8 questions of three marks each.
(v) Section D : Q. No. 35 to 40 comprises of 6 questions of four marks each.
(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 2 questions of one mark each, 2 questions of two marks each, 3 questions of three marks each and 3 questions of four marks each. You have to attempt only one of the choices in such questions.
(vii) In addition to this, separate instructions are given with each section and question, wherever necessary.
(viii) Use of calculators is not permitted.
(i) This question paper comprises four sections – A, B, C and D. This question paper carries 40 questions. All questions are compulsory:
(ii) Section A : Q. No. 1 to 20 comprises of 20 questions of one mark each.
(iii) Section B : Q. No. 21 to 26 comprises of 6 questions of two marks each.
(iv) Section C: Q. No. 27 to 34 comprises of 8 questions of three marks each.
(v) Section D : Q. No. 35 to 40 comprises of 6 questions of four marks each.
(vi) There is no overall choice in the question paper. However, an internal choice has been provided in 2 questions of one mark each, 2 questions of two marks each, 3 questions of three marks each and 3 questions of four marks each. You have to attempt only one of the choices in such questions.
(vii) In addition to this, separate instructions are given with each section and question, wherever necessary.
(viii) Use of calculators is not permitted.
 Question 1
The value of k for which the system of equations x + y – 4 = 0 and 2x + ky = 3, has no solution, is
(a) $2$
(b) $\ne $2
(c) 3
(d) 2 VIEW SOLUTION
 Question 2
The HCF and the LCM of 12, 21, 15 respectively are
(a) 3, 140
(b) 12, 420
(c) 3, 420
(d) 420, 3 VIEW SOLUTION
 Question 3
The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an AP, is
(a) 6
(b) $$6
(c) 18
(d) $$18 VIEW SOLUTION
 Question 4
The first term of an AP is p and the common difference is q, then its 10^{th} term is
(a) q + 9p
(b) p – 9p
(c) p + 9q
(d) 2p + 9q VIEW SOLUTION
 Question 5
If one of the zeroes of the quadratic polynomial x^{2} + 3x + k is 2, then the value of k is
(a) 10
(b) –10
(c) –7
(d) –2 VIEW SOLUTION
 Question 6
 Question 7
The quadratic polynomial, the sum of whose zeroes is –5 and their product is 6, is
(a) x^{2} + 5x + 6
(b) x^{2} – 5x + 6
(c) x^{2} – 5x – 6
(d) –x^{2 }+ 5x + 6 VIEW SOLUTION
 Question 8
The value of p, for which the points A(3, 1), B(5, p) and C(7, –5) are collinear, is
(a) –2
(b) 2
(c) –1
(d) 1 VIEW SOLUTION
 Question 9
The distance between the points (a cos θ + b sin θ, 0) and (0, a sin θ – b cos θ), is
(a) ${a}^{2}+{b}^{2}$
(b) ${a}^{2}{b}^{2}$
(c) $\sqrt{{a}^{2}+{b}^{2}}$
(d) $\sqrt{{a}^{2}{b}^{2}}$ VIEW SOLUTION
 Question 10
If the point P(k, 0) divides the line segment joining the points A(2, –2) and B(–7, 4) in the ratio 1 : 2, then the value of k is
(a) 1
(b) 2
(c) –2
(d) –1 VIEW SOLUTION
 Question 11
Fill in the blank.
Given ΔABC ~ ΔPQR, if $\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{1}{3},\mathrm{then}\frac{\mathrm{ar}\left(\mathrm{\Delta ABC}\right)}{\mathrm{ar}\left(\mathrm{\Delta PQR}\right)}=\mathrm{\_\_\_\_\_\_\_\_\_}.$ VIEW SOLUTION
 Question 12
Fill in the blank.
In the given figure ∆ABC is circumscribing a circle, the length of BC is _____ cm.
VIEW SOLUTION
 Question 13
Fill in the blank.
The value of $\left({\mathrm{sin}}^{2}\mathrm{\theta}+\frac{1}{1+{\mathrm{tan}}^{2}\mathrm{\theta}}\right)$ = ____________.OR
Fill in the blank.
The value of (1 + tan^{2} θ) (1 – sin θ) (1 + sin θ) = _____________. VIEW SOLUTION
 Question 14
Fill in the blank.
A ladder 10 m long reaches a window 8 m above the ground. The distance of the foot of the ladder from the base of the wall is __________ m. VIEW SOLUTION
 Question 15
Fill in the blank.
$\frac{2\mathrm{cos}67\xb0}{\mathrm{sin}23\xb0}\frac{\mathrm{tan}40\xb0}{\mathrm{cot}50\xb0}\mathrm{cos}0\xb0=\mathrm{\_\_\_\_\_\_\_\_\_}.$ VIEW SOLUTION
 Question 16
If the mean of the first n natural number is 15, then find n. VIEW SOLUTION
 Question 17
A die is thrown once. What is the probability of getting a number less than 3?
OR
If the probability of winning a game is 0.07, what is the probability of losing it? VIEW SOLUTION
 Question 18
The ratio of the length of a vertical rod and the length of its shadow is $1:\sqrt{3}.$ Find the angle of elevation of the sun at that moment? VIEW SOLUTION
 Question 19
Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. What is the ratio of their volumes? VIEW SOLUTION
 Question 20
A pair of dice is thrown once. What is the probability of getting a doublet? VIEW SOLUTION
 Question 21
In the given Figure, DE  AC and DC  AP. Prove that $\frac{\mathrm{BE}}{\mathrm{EC}}=\frac{\mathrm{BC}}{\mathrm{CP}}$
OR
In the given Figure, two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2 ∠OPQ.
VIEW SOLUTION
 Question 22
The rod AC of a TV disc antenna is fixed at right angles to the wall AB and a rod CD is supporting the disc as shown in the given figure. If AC = 1.5 m long and CD = 3 m, find (i) tanθ (ii) secθ + cosecθ
VIEW SOLUTION
 Question 23
If a number x is chosen at random from the numbers –3, –2, –1, 0, 1, 2, 3. What is probability that x^{2} ≤ 4? VIEW SOLUTION
 Question 24
Find the mean of the following distribution:
Class: 3 – 5 5 – 7 7 – 9 9 – 11 11 – 13 Frequency: 5 10 10 7 8 OR
Find the mode of the following data :
Class: 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120 120 – 140 Frequency: 6 8 10 12 6 5 3
 Question 25
The minute hand of a clock is 12 cm long. Find the area of the face of the clock described by the minute hand in 35 minutes. VIEW SOLUTION
 Question 26
The sum of the first 7 terms of an AP is 63 and that of its next 7 terms is 161. Find the AP. VIEW SOLUTION
 Question 27
Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are given by 2y – x = 8, 5y – x = 14 and y – 2x = 1.
OR
If 4 is a zero of the cubic polynomial x^{3} – 3x^{2} – 10x + 24, find its other two zeroes. VIEW SOLUTION
 Question 28
Find the area of triangle PQR formed by the points P(–5, 7), Q(–4, –5) and R (4, 5).
OR
If the point C(–1, 2) divides internally the line segment joining A(2, 5) and B(x, y) in the ratio 3 : 4, find the coordinates of B. VIEW SOLUTION
 Question 29
Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial f(x) = ax^{2} + bx + c, a ≠ 0, c ≠ 0.
OR
Divide the polynomial f(x) = 3x^{2} – x^{3} – 3x + 5 by the polynomial g(x) = x – 1 – x^{2} and verify the division algorithm. VIEW SOLUTION
 Question 30
In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then prove that the angle opposite to the first side is a right angle. VIEW SOLUTION
 Question 31
A cone of base radius 4 cm is divided into two parts by drawing a plane through the midpoints of its height and parallel to its base. Compare the volume of the two parts. VIEW SOLUTION
 Question 32
A man can row a boat downstream 20 km in 2 hours and upstream 4 km in 2 hours. Find his speed of rowing in still water. Also find the speed of the stream. VIEW SOLUTION
 Question 33
In the given figure 5, two circles touch each other at the point C. Prove that the common tangent to the circles at C, bisects the common tangent at P and Q.
VIEW SOLUTION
 Question 34
Prove that : $\frac{\mathrm{cot}\mathrm{\theta}+\mathrm{cosec}\mathrm{\theta}1}{\mathrm{cot}\mathrm{\theta}\mathrm{cosec}\mathrm{\theta}+1}=\frac{1+\mathrm{cos}\mathrm{\theta}}{\mathrm{sin}\mathrm{\theta}}$ VIEW SOLUTION
 Question 35
The following table gives production yield per hectare (in quintals) of wheat of 100 farms of a village :
Production yield/hect. 40 – 45 45 – 50 50 – 55 55 – 60 60 – 65 65 – 70 No. of farms 4 6 16 20 30 24
Change the distribution to 'a more than' type distribution and draw its ogive.OR
The median of the following data is 525. Find the values of x and y, if total frequency is 100:
Class : Frequency: 0 – 100 2 100 – 200 5 200 – 300 x 300 – 400 12 400 – 500 17 500 – 600 20 600 – 700 y 700 – 800 9 800 – 900 7 900 – 1000 4
 Question 36
A bucket in the form of a frustum of a cone of height 30 cm with radii of its lower and upper ends as 10 cm and 20 cm, respectively. Find the capacity of the bucket. Also find the cost of milk which can completely fill the bucket at the rate of Rs. 40 per litre. $\left(\mathrm{Use}\pi =\frac{22}{7}\right)$ VIEW SOLUTION
 Question 37
Show that the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.
OR
Prove that one of every three consecutive positive integers is divisible by 3. VIEW SOLUTION
 Question 38
The sum of four consecutive numbers in AP is 32 and the ratio of the product of the first and last terms to the product of two middle terms is 7 : 15. Find the numbers.
OR
Solve : 1 + 4 + 7 + 10 + ... + x = 287 VIEW SOLUTION
 Question 39
Draw a ΔABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of ΔABC. VIEW SOLUTION
 Question 40
From the top of a 7 m high building the angle of elevation of the top of a tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. VIEW SOLUTION
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